# Pahingi naman po ng sampol na Solution sa ARITHMETIC,GEOMETRIC,HARMONIA and Fibonacci !

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For example, consider 4, 7,10.... In this sequence, first term is 4 and we get the second term as 7 = 4 + 3. Second term is 7 and we get third term as 10 = 7 + 3. So, it is clear that the difference between the two successive terms is 3 and it is the same for all terms or in the complete sequence. So, this is the example for arithmetic sequence.

Consider the sequence 2, 7, 12, 17, 22,.......

1. The nth term of any arithmetic sequence is given by Tn = a + (n - 1)d

2. The 6th term(taking n = 6), T6 = 2 + ( 6 -1) 5 = 27

3. The 8th term(taking n = 8), T8 = 2 + ( 8 -1) 5 = 37Fibonacci Sequence

In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. has no visible pattern. But the sequence is generated by the recurrence relation and is given by

a1 = a2 = 1

a3 = a1 + a2

an = an - 2 + an - 1, n > 2

Arithmetic SequenceIf we have a sequence as x1, x2 ,x3,......., and the sequence follows the rule as xj - xi = d for all values of i and j, then the sequence is called the arithmetic sequence. This type of sequence where any term except the first term is obtained by adding a fixed number to the previous term.

If we have x1, x2 , x3 ,.....,xn,..... be the any arithmetic sequence, then the sum of this finite series, where n is any positive integer is given by x1+x2+..............+xn=∑ni=1xi

Geometric Sequence

A sequence where any term except the first term is obtained by multiplying a fixed number to the previous term is called geometric sequence. If the sequence x1, x2, x3,.......,xn follows the condition xn+1xnxn+1xn = r, then that sequence is known as geometric sequence and r is known as the common ration of the sequence.

Geometric Sequence Examples:

Consider the sequence 10, 20, 40, 80

Here, a = 10, r = 2

5th term is T5 = 10 x (2) 5 - 1 = 10 x (2) 4 = 160

7th term is T7 = 10 x (2) 7 - 1 = 10 x (2) 6 = 640

Harmonic Sequence

This is a type of sequence where the reciprocal of the terms from an arithmetic sequence. So, if any sequence is in the form of 1x1,1x2,1x3,.......,1xn1x1,1x2,1x3,.......,1xn, then the sequence is said to be harmonic sequence.

If a, b and c are in the form of arithmetic sequence, then b = 2aca+c2aca+c

For example, the sequence 11,12,13,.....11,12,13,..... is a harmonic sequence.